Warm-Up Example

This is a simple system where one can obtain an explicit expression for the solvation thermodynamic quantities. It is also a warm-up example for the two more complex problems that bear resemblance to the problem of the effect of a solute on the structure of water. The reader is urged to do this exercise before turning to exercises E.3.11 and 3.12. 

Warm-Up Example 3 Convert 50 °C to a temperature in degrees Ranldne. 

After solving a problem, it is highly advisable to test the result in order to understand whether it is an expected and reasonable result and, ideally, to verily whether it is the true solution to the problem. Before moving on to Sect. 5.3, we cordially invite you to solve the following warm-up examples. 

In our view, the difficulty is that when faced with a problem like warm-up example 1, most people are wrongly tempted to analyze and solve it mentally and not take the problem seriously. The strong and potent message here is that one should always crmsider problems as a worthwhile challenge and solve it using some method. As we have shown, if you analyze all the possibiUties (just three), it is easy to see that it is better to change doors ... 

First, for now, these nine points are simply a list of recommendations, but we will analyze these recommendations later when we focus on developing an integral and general procedure for approaching, formulating, and solving material balance problems. Second, we will exemplify the relevance of these tips on warm-up examples and when solving problems in Sect. 7.9 (solved... 

As depicted in Fig. 7.15 the proposed procedure consists of five steps in series that are described and analyzed below. Later, in the warm-up examples, we will apply it to problems. 

Additional considerations on the degrees of freedom. We already have all the tools to determine the values for NV, NMB, NSV and NK, then we can determine DF. Now we need to understand that its use has some limitations. For exampie if we have some degrees on freedom in your process, then, certainly you will be able to fix some unknown variables in your process but with some limitations. For example if a process-unit has three degrees of freedom and with one stream at the input and two streams at the output you cannot fix ail three streams arbitrarily, there is a mass balance equation that relates these three streams, then, in this case, you can fix two of these three streams. We are not implying that you cannot use all the DF, just to state that some constraints limit your freedom in the choices of values and variables used as DF. This concept wili be further analyzed and exemplified in tiie warm-up examples and reinforced in section 7.9. For example in warm-up example 2 we will show how to arralyze the degrees of freedom in a multiple unit process including a side-stream. 

As wiil be shown in warm-up example 2 if the process includes by-passes, side-streams, and/or recycle streams (see Figure 7.16) it is advisable to include an additional process-unit (mixer) in your flow-diagram showing the mixture of one stream and, for example, a recycle stream. This advice is critical because you need to consider this "additional unit" when determining the number of independent materials balance equations NMB). in addition, if a stream is divided (e.g. side-stream), all new streams will have the same composition but you need to add one total material balance equation. When the side-stream joins the process again, it is necessary to add a new process-unit (mixer). 

Once you have the results is necessary to check if they are according to your expectations and more than that, hopefuily test if they are correct. In warm-up examples 1 and 2 we will show you some cases were results are properly tested. In addition, this concept will be further analyzed in section 7.9 (solved problems). 

As Einstein said, Example is not another way to teach, it is the only way to teach." Because we agree with this statement, we are fully committed to giving you a large number of examples, either solved or to be solved. In this section we will provide some warm-up examples to start practicing the proposed procedure. 

It is definitely advisable to follow these two warm-up examples step by step. They are simple but will guide you in the first steps of the proposed procedure. 

NOTE As we will see in warm-up example 2, it is not necessary here to use two variables for mass fraction composition in each stream. For example, we know thatXH2oi = 1 - si. so we can reduce the total number of variables used here to just one mass fraction as a variable, fii this simple example, it is not really critical, but in general we will try to minimize the number of variables ... 

Warm-Up Example 1. Acetylene is a gas that can be obtained by treating calcium carbide with H2O according to the following reaction ... 

Warm-Up Example 2. What happens if you are feeding, for example, 6 [mol] of A and 3 [mol] of B in reaction (8.12) Then reagent A is in excess (more than the required amount) and, in addition, we can say that reagent B is limiting. As seen in this example, we can deduce... 

In warm-up example 2, we stated that compound A was in excess, but can we quantify its excess Yes. How By defining the fractional excess as... 

Then, in warm-up example 2, we can calculate the excess of reagent A. 

As has been stressed throughout the book, and particularly in the previous chapter, a correct procedure is, again, essential to solving material balance problems for reactive systems. Warm-up examples 4 and 5 focus on managing material balances for reactors. 

Warm-Up Example 4. A reactor is fed with a mixture of oxygen (O2) and propane (CsHg). Oxygen is in excess and the conversion of the reaction is 70 %. At the output of the reactor, the presence of 21 [mol] of CO2 and 40 [mol] of O2 has been analytically determined, (a) What is the fractional excess of oxygen (b) How many moles of propane were fed to the reactor (c) What is the molar composition of the output stream ... 

As shown in warm-up example 4, a practical way to deal with material balances for a reactor is to construct a table (Table 8.3). From now on, all the material balances for reactors will be done using this procedure. 

Warm-up example 6. A continuous stirred-tank reactor (CSTR) of volume V is fed with a stream of F L/h and with a substrate concentration of So [kg/L], The transformation speed of the substrate within the reactor is directly proportional to its concentratitMi (—kS), where k has units of l/h. If we can assume that the concentration of the substrate at the output stream is equal to the concentration within the reactor, and the reactor is operated under steady state (Fig. 8.4), then express the substrate concentration at the output stream (S) as a function of known variables (F, V, k, and 5o). 

Warm-up example 7. A clever young chemical engineering student is proposing that you analyze the previous problem, but this time instead of using one reactor of volume V, he is suggesting using two reactors of volume VH each. He is clever, but not as well prepared as you, and so he asks you to investigate two alternatives, to maximize the disappearance of the substrate, as follows (a) Use both reactors in parallel, (b) Use both reactors in series, (c) What would your recommendation be and... 

Corollary In warm-up examples 4 and 5 (chemical reactors), we had information about stoichiometry and conversion, and the proposed procedure was to construct a table to take into account the moles entering the reactor, moles reacting, and the moles leaving the reactors (reagents and products). This is a convenient procedure and facilitates the material balance in the reactor. On the other hand, in the bioreactor problems, we had information about the disappearance of the substrate (kinetics), and in that case it was easier just to formulate the mass balance like (8.6). 

As was mentioned, we will follow the same strategy that was developed for material balance for nonreactive systems. Some slight differences are that here we will use moles instead of mass, and the material balance through the reactor will be done developing the table that was presented in warm-up example 4. 

Continuous steady-state bioreactors [6]. In warm-up example 7, we discovered that for this specific example, it was a better option to arrange two bioreactors of volume V/2 in series instead of one bioreactor of volume V. A company interested in your skUls wants to design a system of bioreactors with a total volume of 500 [L] and get an output substrate concentration lower than 31 [g/L]. 

Now we will show you how to approach this warm-up example problem with both procedures as follows ... 

The following image shows the first step for the warm-up example (Fig. 11.9). 

In this section we present and develop three maximum and minimum problems and two OR problems. In both cases the emphasis will be on problem formulatiOTi because in the warm-up examples we already detailed the solution procedure with the help of the Excel spreadsheet. 

As suggested in the warm-up example, we will first plot a graph to find the shape of the curve and a tentative solution. Then we will use the Solver tool to confirm the graphic solution or get a more accurate solution. 

As explained in the warm-up example, we first choose a cell for v (D9) and assign it a value of 0. Then we write the function in cell G9. Now click on Data and then Solver. In the Solver tool box we set the objective function (G9), choose minimum, Min, then in By Changing Variable Cells, we select D9 and finally click on Solve to get (Fig. 11.26)... 

As detailed and explained in the warm-up example (11.5.5) we will use the Solver tool from Microsoft Excel. The following screen shows the objective function (cell L6), the variables R and P (cells 19 and no, respectively), and the constraints for Flour, Capital and Hundredweights (cells D9, DIO and D11 respectively). We tentatively start with initial values of R = 50 and P = 30 (Fig. 11.35). 

---